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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, C06017, doi:10.1029/2005JC003155, 2006

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Morphodynamic evolution of double-barred beaches M. D. Klein1,2 and H. M. Schuttelaars3,4 Received 8 July 2005; revised 13 February 2006; accepted 14 March 2006; published 13 June 2006.

[1] The linear and nonlinear morphological behavior of double-barred coastal systems

under the forcing of obliquely incident waves is studied using a nonlinear numerical model. The linearly most unstable bed forms consist of crescentic patterns (rip channels), whose spacing depends on the magnitude of the longshore current velocity. Using the nonlinear model, six morphodynamic experiments have been performed with various initial bed perturbations in order to assess, among others, the influence of the initial bed perturbation on the morphodynamic evolution. The nonlinear experiments have been pursued well into the nonlinear regime, showing that after a phase of initial exponential growth, a highly dynamic behavior is observed and no equilibrium state is reached. The spacings predicted with the linear stability analysis are observed during the exponential phase of the nonlinear experiments. In the dynamic phase, however, four to seven modes significantly contribute to the resulting bed features. In this final stage, the apparent wavelength of 1000 m of the resulting bed forms on the inner bar is quite insensitive to the initial bed perturbation. On the outer bar it seems that the longer the wavelength of the initial bed perturbation, the longer the wavelength of the final bed forms in the dynamic phase and the larger the migration celerity. In general, the bed forms can be characterized as crescentic or undulating bed patterns. Good correspondence between simulated and observed spacings, shapes and migration celerities are found. Citation: Klein, M. D., and H. M. Schuttelaars (2006), Morphodynamic evolution of double-barred beaches, J. Geophys. Res., 111, C06017, doi:10.1029/2005JC003155.

1. Introduction [2] The cross-shore profile of most uninterrupted, sandy coastal systems can be characterized as a planar sloping, single-barred or double-barred profile. Examples of doublebarred systems are the Duck coast (North Carolina) and the Holland coast (the Netherlands). In the case of the doublebarred Dutch coast Ruessink and Kroon [1994] and Wijnberg and Terwindt [1995] have shown that the two breaker bars and the swash bar of that system exhibit a cyclic behavior. One cycle consists of the generation of a (swash) bar near the coast, followed by an offshore migration and, finally, the degeneration of that bar. Depending on the exact location along the Dutch coast, the cycle period is 4 to 15 years. [3] Apart from this cyclic behavior of the shore-parallel bars, 3D morphological features are superposed on these bars. These features consist of rip channel systems, crescentic, irregular and undulating bed forms with length scales ranging from a few tens of meters to a few thousands of meters [see, e.g., Bowen and Inman, 1971; Wright and Short, 1984; Konicki and Holman, 2000; Van Enckevort and Ruessink, 2003]. An extensive overview of observa1 Section of Hydraulic Engineering, Delft University of Technology, Delft, Netherlands. 2 Presently at Svasˇek Hydraulics, Rotterdam, Netherlands. 3 Faculty of Geosciences, Utrecht University, Utrecht, Netherlands. 4 Now at Delft Institute of Applied Mathematics, Delft University of Technology, Delft, Netherlands.

Copyright 2006 by the American Geophysical Union. 0148-0227/06/2005JC003155$09.00

tions of crescentic bed forms is given by Van Enckevort et al. [2004]. Along the Dutch coast, these 3D features can be very persistent but typical time scales at which these features change are of the order of days. This is much smaller than the time scale at which the shore-parallel bars (2DV morphology) change significantly. [4] Two different theories have been formulated to explain rhythmic features in the surf zone: The first one attributes these features to direct hydrodynamic forcing [see, e.g., Bowen and Inman, 1971; Holman and Bowen, 1982] and the second theory considers self-organization in the coupled hydro- and morphodynamic system. The occurrence of rhythmic features whose length scales do not appear in the forcing, like mega ripples, favors the explanation of self organization. Often the two mechanisms are studied in separation, although Reniers et al. [2004] have presented a model that incorporates both template forcing and self-organization. [5] The initial stage of self-organization can be studied with linear stability analyses (LSAs), describing the initial growth of bed perturbations from an otherwise longshoreuniform coastal system. The linear stability of planar beaches is studied by, among others, Hino [1974], Christensen et al. [1994], Falque´s et al. [1996, 2000], Ribas et al. [2003] and Klein and Schuttelaars [2005], showing that the linear stability characteristics, i.e., the preferred wavelength and associated growth and migration rates, are very sensitive to model formulations, and especially to the sediment transport formula used. [6] Deigaard et al. [1999], Calvete et al. [2002], Coco et al. [2002], Damgaard et al. [2002] and Caballeria et al. [2003]

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Figure 1. Definition sketch of the equilibrium state. For a discussion of the symbols, see the text.

focus on the linear stability of single-barred beaches. These studies conclude that longshore-uniform barred coastal systems are unstable under the forcing of breaking waves. On the crests of the breaker bars, bed perturbations grow with a spatial structure corresponding to rip channel systems, characterized by alternating crescentic shaped shoals and channels around the crest of the bar and by a certain preferred longshore spacing. The variation in the preferred spacing found in these studies is rather large, namely, 2 to 10 times the distance between the shoreline and the crest of the breaker bar. [7] A large variation in the growth rates is found as well. Damgaard et al. [2002] found typical growth rates of O(0.2d
1) Deigaard et al. [1999] of O(2 d
1) and Calvete et al. [2002] O(20 d
1). Despite these large differences in growth rates, most studies find the same dependency of the growth rate on the wave height and the distance between the shore and the crest of the bar, namely, increasing growth rate with increasing wave height and with decreasing distance between the shore and the crest of the bar. [8] Damgaard et al. [2002] investigate the finite amplitude behavior of bed perturbations of single-barred beaches as well. They show that the longshore wavelength of the initial bed perturbation is important for the morphological development. If that wavelength is small, in their case 93 m, only growth of that mode is observed. For larger initial wavelengths, other modes than the one initially imposed emerge. Furthermore, they find good correspondence between the characteristic length scale found in the linear and the nonlinear regime. [9] The growth of rhythmic features on double-barred beaches has not been investigated in detail before and is the focus of this paper. Both the linear stability and the subsequent nonlinear evolution is studied. The sensitivity of the linear stability characteristics to the geometric configuration of the double-barred system and to the wave height is investigated. The linear stability analysis is performed using a state-of-the-art numerical model. Hence

no process simplifications have to be made. Another advantage of this approach is that the bed evolution is easily extended into the nonlinear regime using the same model. By performing these morphodynamic computations a number of questions will be answered. First of all, the time interval over which the morphological development of a coastal system can be regarded as linear is determined. Secondly, it is assessed whether the length scale of the fastest growing mode as found in the linear stability analysis still dominates when nonlinear processes become important. Furthermore, the existence of a morphodynamic equilibrium and the time needed to reach this equilibrium are investigated. [10] This paper is organized as follows. Section 2 discusses the numerical model that has been used. The method of performing LSAs and the nonlinear morphodynamic experiments is discussed in section 3. The results of the linear stability analysis are presented in section 4 and the results of the nonlinear experiments in section 5. An overall discussion can be found in section 6 and the conclusions are drawn in section 7.

2. Model Formulations [11] The coastal system under study is a double-barred beach system. Initially, the coastal system is uniform in the longshore direction, see Figures 1 and 2 in which the coordinate system and a number of variables are defined. Here the cross-shore (long-shore) coordinate is denoted by x (y) and z denotes the vertical one. [12] The equilibrium bed level with respect to the still water level is denoted by h0(x). The position of the bar crest closer to (further from) the beach is defined by zb1 = h0(wb1) (zb2 = h0(wb2)), and the position of the trough closer to (further from) the beach by zt1 = h0(wt1) (zt2 = h0(wt2)). The still water depth at x = 0 is set to 1 m in order to exclude significant bed changes at the coastline from this study.

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[15] The water motion is only forced near the coast by obliquely incident, breaking waves with a significant wave height Hs and an angle of wave incidence q prescribed at the seaward boundary, resulting in a wave setup and a longshore current. The propagation of these obliquely incident, irregular short waves and their refraction and breaking is calculated using the 2nd generation wave model HISWA [Holthuijsen et al., 1989] which is based on an evolution equation of the wave action density spectrum, @Ncg;x @Ncg;y @Ncg;q þ þ ¼
Dw ; @x @y @q

Figure 2. Definition sketch of the geometrical parameters. For a discussion of the symbols, see the text. [13] The water motion is described by the depth- and wave-averaged shallow water equations [Phillips, 1977; Horikawa, 1999], @h @uc D @vc D þ þ ¼0 @t @x @y

@uc @uc @uc @h tx þ uc þ vc þg þ @t @x @y @x rw D
2  Fx @ uc @ 2 uc
þ
n ¼0 rw D @x2 @y2

ty @vc @vc @vc @h þ uc þ vc þg þ @t @x @y @y rw D
2  Fy @ vc @ 2 vc
þ 2 ¼ 0:
n rw D @x2 @y

ð1Þ

in which the wave action N [m2sHz
1] is defined as the variance density E [m2Hz
1] divided by the wave frequency f. Furthermore, cg,x, cg,y [ms
1] and cg,q [rads
1] are the propagation speeds of wave action in the x, y and angular space, and q is the wave angle. The wave action dissipation Dw [m2s1] is computed with the breaker model of Battjes and Janssen [1978]. This dissipation is used to compute the wave forces ~ F per surface area with the formulation proposed by Dingemans et al. [1987], ~ F ¼ rw gDw f~ k;

Here t denotes time, h the free surface elevation, uc (vc) the cross-shore (longshore) current velocity, D the total water depth, g the acceleration of gravity, tx and ty the bed shear stresses, Fx and Fy the wave forces per surface area, rw the water density and n a uniform turbulent eddy viscosity. The effects of currents and waves on the direction and the magnitude of the bed shear stresses are taken into account by means of the parameterization of Soulsby et al. [1993] of the bed shear stress formulation proposed by Fredsøe [1984]. [14] At the seaward boundary of the model a zero water level boundary condition is applied. At the beach the boundary condition consists of a zero cross-shore velocity and a free-slip condition. The Delft3D modeling system [see Roelvink and van Banning, 1994] has been used to solve the governing equations. To solve these equations numerically, only a finite extent in the longshore direction can be studied. On these lateral boundaries, boundary conditions have to be prescribed in such a way that an infinite system is approached as close as possible. The applied lateral boundary conditions are discussed in sections 3.1 and 3.2.

tan ~ bs k~ uk ~ k~ u k3 u
tan fi

fcw b ¼ gðs
1Þ tan fi fcw s þ gðs
1Þws

ð3Þ

ð5Þ

with ~ k being the wave number vector. Wave energy not dissipated in the surf zone does not reflect at the beach but leaves the model domain. qi is [16] A wave-averaged sediment transport vector h~ derived by averaging, denoted by hi, the formulations of Bailard [1981] over the wave period. The Bailard [1981] formulations for bed and suspended load read ~ qs q ¼~ qb þ ~

ð2Þ

ð4Þ

!

2

k~ u k3 ~ u


! s tan ~ bs 5 ~ kuk ; ws

ð6Þ

in which fcw is a friction factor taking into account the friction due to waves and currents, ~ u the total velocity vector, b (s) the efficiency factor for bed (suspended) load, s the relative density rs/rw with rs the density of the sediment, fi the angle bs the bed slope of repose, ws the sediment fall velocity and ~ vector. The wave-averaged version of Bailard’s formulation u0sin(ft) in equation (6) is deduced by substituting ~ u =~ uc + ~ with f the (peak) frequency of the short wave, ~ uc = (uc, vc) and ~ u0 the wave orbital velocity. After averaging over the wave period 2p/f, the wave-averaged bed and suspended qsi read load terms h~ qbi and h~  1 1 uc u2c þ v2c þ u20 þ v20 ~ 2 2 #  E tan ~ bs D 2 2 3=2 þðuc u0 þ vc v0 Þ~ u0
u þv tan fi

hq~bi ¼

fcw b gðs
1Þ tan fi




ð7Þ

" # E D E  tan ~ fcw s bs D 2 s 2 2 3=2 2 5=2 ðu þ v Þ ~ : ðu þ v Þ u
hq~s i ¼ gðs
1Þws ws ð8Þ

Where possible, the averaging has been done explicitly, but for the last term of equation (7) and for equation (8) the

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averaging has to be done numerically. More details are given by Van der Molen [2002] [see also Klein and Schuttelaars, 2005]. [17] Bed changes, finally, are computed with the sediment mass conservation equation, ð1
pÞ

@h ~  h~ ¼
r qi; @t

ð9Þ

in which p is the bed porosity.

3. Method of Analysis [18] The above described system of equations allows for a longshore-uniform morphodynamic equilibrium; see Klein and Schuttelaars [2005] for a detailed discussion. The stability of this morphodynamic equilibrium will be investigated using a linear stability analysis (section 3.1) and by studying its temporal evolution. The method to study this morphodynamic evolution is described in section 3.2. 3.1. Linear Stability Analysis [19] The linear stability of the longshore-uniform morphodynamic equilibrium is assessed by performing a LSA. In this study a fully nonlinear numerical model is used to calculate the wave forcing, the water motion and the sediment transport rates. Therefore a formal linearization of the equations can not be made and the linear stability characteristics of double-barred beaches have to be investigated using a different method. The method used is extensively discussed by Deigaard et al. [1999] and Klein and Schuttelaars [2005] and only a short description of the method is given here. [20] An iteration process is used to find the solution with the largest growth rate, irrespective of whether this largest growth rate is positive or negative. An initial guess is made for the bed perturbation. This is done by choosing a crossshore amplitude function b(x) with a maximum amplitude of 0.01 m and a longshore wavelength l. This wavelength is kept fixed throughout the iteration process. This perturbation is added to the longshore-uniform reference bed. Using this topography the water motion, sediment transport rates and bed changes are computed. A Fourier decomposition is used to retrieve the bed changes associated with the wavelength under consideration. The resulting cross-shore amplitude function of these bed changes is denoted as L(x). The initial amplitude function and the newly computed one are used to make a guess for the eigenvalues via the Rayleigh quotient R [see Griffel, 1985], R1 LðxÞb*ðxÞdx R ¼ R01 ¼
iw; * 0 bðxÞb ðxÞdx

ð10Þ

in which * denotes the complex conjugate and w = wr + iwi the complex eigenvalue. [21] The next iteration starts with the newly computed bed perturbation and the above-described process is repeated until the bed perturbation and the growth and migration rates do not change from one iteration to the next. The criterion for convergence is that both the growth and migration rate change less than 0.5%. If the process has converged, the resulting bed perturbation is the eigenfunc-

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tion and the associated growth and migration rates are the imaginary and the real part of the eigenvalue, respectively. The migration celerity is obtained by dividing the migration rate by the longshore wavenumber. Repeating this iteration process for a range of wavelengths yields the typical growth rate versus wavelength graphs from which the solution with the largest growth rate can be identified. This solution is referred to as the fastest growing mode (FGM). The wavelength, growth and migration rate associated with the FGM are indexed with ‘‘p.’’ [22] When applying the numerical model to perform LSAs periodic longshore current velocities, resulting from the previous iteration, are imposed on the lateral boundaries. Since the current velocities obtained with the previous iteration do not exactly match the bed perturbations of the present iteration, small discrepancies exist between the actual longshore current pattern and the ones imposed on the boundary. Therefore the Fourier decomposition is only applied in a domain well away from the lateral boundaries. This sets the upper limit for the wavelength that can be assessed with the numerical model, namely, 3000 m, although the model domain is 6000 m. The lower limit is 300 m, as the above described iteration process does not converge for wavelengths smaller than approximately 300 m. 3.2. Nonlinear Experiments [23] The nonlinear morphological behavior of doublebarred beaches is explored by means of morphodynamic experiments in which the bed perturbations are allowed to grow in time. A bed perturbation is defined as the difference between the actual topography and the initial (longshoreuniform) topography. In the model setup of the nonlinear analysis the model domain has been extended from 6 km to 12 km, because the bed perturbations become large and consequently the disturbances from the model boundary will be larger and more extended. As boundary conditions on the lateral boundaries, the longshore current velocity profiles associated with the basic state have been imposed. [24] In general, the morphodynamic experiments are performed by perturbing a longshore-uniform coast with an initial bed perturbation. This initial perturbation has been constructed such that its maximum amplitude is everywhere equal to or smaller than 0.01 m. With this bathymetry the numerical model is run to obtain the stationary, wave-driven flow field. This flow field yields the sediment transport rates resulting in bed changes. The new bathymetry is computed using a morphological time step chosen such that the bed changes are at most Dhmax m per time step. Using this new bathymetry the numerical model is run again, etc. 3.3. Bed Slope Related Sediment Transport [25] As has been shown by Klein and Schuttelaars [2005], bed slope terms have, for a realistic range of parameter values, a negligible influence on the linear stability characteristics. In the LSAs, bed slope related sediment transport has therefore not been included. [26] In order to concur with the linear experiments, the first four nonlinear experiments (NL1, NL2, NL3 and NL4) have been performed without bed slope terms as well. Next, two experiments (NL5 and NL6) are performed, including bed slope related sediment transport.

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Table 1. Summarized Settings of the Nonlinear Experiments Exp.

Applied Settings

Initial Bed Perturbation

LFD, m

NL1 NL2 NL3 NL4 NL5 NL5B NL6

L1 L4 L4 L4 L1 L1 L4

l = 600 m l = 2200 m l = 700 m l = 500, 700, 900, 1400 and 2000 m random random random

6000 6600 6300 6000 6000 6000 6000

[27] Preliminary experiments with and without bed slope related sediment transport showed that perturbations with small wavelengths ( 30 m) in the longshore direction finally dominate the bathymetry. These bed forms developed at the outer bar, although only a very weak longshore current was present there. These bed forms are considered unrealistic, because the grid cell size is 10 m. To prevent these physically unrealistic oscillations, a diffusive term proportional to hyy has been added to the right-hand side of equation (9), such that the linear growth rates of the bed forms with wavelengths of interest (between 300 and 3000 m) change less than 1%, whereas smaller wavelengths are sufficiently damped to not dominate the bed evolution.

4. Results of the Linear Stability Analysis: Initial Evolution 4.1. Definition of the Linear Experiments [28] In Table 1 the parameter values of the double-barred reference experiment are given. These parameters are representative of the Dutch coastal system. As already discussed in section 1, the shore-parallel bars of the Dutch coastal system exhibit periodic behavior [see Ruessink and Kroon, 1994; Wijnberg and Terwindt, 1995]. The geometrical parameters presented in Table 1 represent a stage in the morphodynamic cycle in which the outer bar is relatively low. Therefore, to Table 2. Default Parameter Values Used in the Numerical Experiments Parameter Length model domain (LSA) Length model domain Width model domain Cross-shore grid size Longshore grid size Vert. position inner trough Vert. position inner bar Vert. position outer trough Vert. position inner bar Hor. position inner trough Hor. position inner bar Hor. position outer trough Hor. position outer bar Bed level at x = 0 Significant wave height Peak period Wave angle Drag coefficient Eddy viscosity Gravitational acceleration Water density Sediment density Median grain size Porosity

Value Ly Ly Lx dx dy zt1 zb1 zt2 zb2 wt1 wb1 wt2 wb2 z(0) Hs Tp q cd n g rw rs D50 p

6000 12000 1200 10 10 1.9 1.5 4.5 2.7 70 130 240 340
1 1.1 6 10 0.0035 1 9.81 1030 2650 250 0.4

Grid Cell Size 10 10 10 10 20 20 20





10 10 10 10 20 20 20

m2 m2 m2 m2 m2 m2 m2

Bed Slopes Effects Included? no no no no yes no yes

study the influence of the geometry on the linear stability characteristics, the height of the outer bar is varied while all other geometrical parameters are kept fixed. [29] Apart from the variations in the bar profile, the wave characteristics vary significantly throughout a year as well. A significant wave height of 1.1 m, representative of the yearlyaveraged wave height observed along the Dutch coast [Van Rijn, 1997], is imposed at the seaward model boundary in the reference experiment. The imposed wave angle is 10 at the offshore boundary, corresponding to a wave angle of approximately 5 at the crest of the outer breaker bar. By increasing the wave height at the seaward boundary to 3 m, the influence of storm conditions on the linear stability characteristics is investigated. The wave angle at the offshore boundary is fixed at 10 . [ 30 ] The reference experiment (L1) is discussed in section 4.2. The linear stability characteristics resulting from varying only the bar height (experiment L2) and only the wave height (experiment L3) are presented in section 4.3. Experiment L4, in which both the height of the bar and the wave height have been varied, is discussed in section 4.4. Table 2 summarizes the parameter values used in the various experiments. 4.2. Reference Experiment [31] Figure 3 shows the growth rate of bed perturbations on top of the double-barred profile for different longshore wavelengths, using the default parameter values (Table 1). All wavelengths considered have positive growth rates and a maximum growth rate occurs for a wavelength lp of 600 m. This mode is designated as the fastest growing mode (FGM).

Unit m m m m m m m m m m m m m m m s m2s
1 ms
2 kg m
3 kg m
3 mm

Figure 3. Reference experiment: growth rate versus longshore wavelength.

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the inner bar and therefore does not appear in Figure 5. Furthermore, it is observed that the larger the longshore wavelength, the larger the ratio of the amplitude of the bed perturbation on the inner and outer bar.

Figure 4. Reference experiment: migration rate versus longshore wavelength. Except for the FGM, no other (local) maximum is found. The associated growth rate wi,p is 2.84 d
1, corresponding to an e-folding time of 0.35 d. [32] The migration rate of the FGM is positive, meaning down-flow migrating bed features. Scaling with the wave number yields the migration celerity, which is 79.6 m d
1 for the FGM. The migration rate linearly decreases with the longshore wavelength for wavelengths between 500 m and 1200 m, as Figure 4 demonstrates. Surprisingly, the migration is in up-current direction for relatively large wavelengths. [33] The bed perturbation of the FGM is depicted in Figure 5 and can be characterized as a rip channel system with alternating channels and shoals around the crest of the inner bar. Shoals are indicated by solid contours and channels are indicated by dashed contours. In the case of the FGM, the amplitude of the bed perturbation on the outer bar is only 1.3% of the amplitude of the bed perturbation on

Figure 5. Reference experiment: bed and flow perturbations of the FGM. The straight dash-dotted lines indicate the crests of the breaker bars. Solid contours are shoals, and dashed contours are channels.

4.3. Experiments L2 and L3 [34] Figure 6 illustrates the sensitivity of the growth rate to a larger wave height (experiment L2) and a higher outer bar (experiment L3) compared to the reference experiment. Increasing the wave height yields an increase of both the preferred spacing and the growth rate. Increasing the height of the outer bar yields a reduction of the growth rate, whereas the preferred spacing shifts to slightly smaller wavelengths. [35] The bed perturbations resulting from experiments L2 and L3 are qualitatively the same as the ones obtained in the reference experiment. They consist of rip channel systems on both bars whereas the ratio of the maximum amplitude of the bed perturbations on the inner and outer bar depends on the wave height, the height of the outer bar and the longshore wavelength. The larger the wave height, the height of the outer breaker bar and the longshore wavelength, the smaller this ratio is. 4.4. Experiment L4 [36] Experiment L4 is performed with Hs = 3.0 m and zb2 = 2.0 m, thus more energetic conditions and a higher outer bar with respect to the reference experiment. In Figure 7 the growth rate as a function of the longshore wavelength is denoted by the solid line marked with crosses. It clearly demonstrates that this dependence is far more complicated than the ones obtained with experiments L1, L2 and L3. Nonetheless, a FGM has been found for a wavelength of 2200 m. The corresponding growth rate and migration celerity are 3.24 d
1 and 385.2 m d
1, respectively. Besides the occurrence of a global maximum, several local maxima are found as well, namely, for l = 400, 700 and 900 m. [37] Figures 8 and 9 depict two bed and flow perturbations, for l = 600 m and lp = 2200 m, respectively. Significant perturbations develop on both bars although the amplitudes of the bed perturbation on the outer bar

Figure 6. Experiments L1, L2, and L3: growth rate versus longshore wavelength.

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has been checked by performing a short time integration of the fully nonlinear system of equations. The initial bed perturbation consists of the most unstable eigenmode obtained with the LSA. By fitting the time evolution of the amplitude of the eigenmode to an exponential curve, the growth rates are obtained in an independent way and are depicted in Figure 7 by the solid line marked with pluses. Figure 7 shows that not only the qualitative behavior, i.e., the occurrence of a FGM for l = 2200 m and multiple local maxima, but also the values of the growth rates determined with the two methods are virtually the same and are within the range of accuracy required for convergence. Also, the shapes of the bed perturbations are similar to the ones obtained with the LSA. The other linear experiments have been validated with this method as well.

Figure 7. Experiment L4: growth rate as a function of the longshore wavelength, determined with the LSA (crosses) and with a morphodynamic experiment (pluses) with Dhmax = 0.002 m. associated with l = 600 are much smaller than those associated with l = 2200 m. The ratio Aout/Ain of the amplitudes of the bed perturbation on the inner and outer bar for these two experiments is 1.03 and 5.88, respectively. In general, the ratio Aout/Ain increases with increasing wavelength. [38] Furthermore, the locations of the maximum amplitude on both bars are not exactly in phase. Disregarding this small phase shift in longshore direction, the bed perturbation on the inner bar is the mirror image of the bed perturbation on the outer bar: When a shoal at the seaward side of the inner bar is found, a channel is observed for the same longitudinal coordinate at the seaward side of the outer bar. These observations of the bed perturbation hold for all wavelengths. [39] Since the linear stability curve obtained in this experiment is rather unexpected, the linear stability analysis

Figure 8. Experiment L4: bed and flow perturbations associated with l = 600 m.

5. Results of the Nonlinear Experiments: Nonlinear Evolution [40] In this section the nonlinear morphodynamic evolution of double-barred beaches is investigated. To do so, a choice for the maximum allowable bed change Dhmax has to be made; see section 3.2. Two contradictory considerations are of importance for the choice of Dhmax. On the one hand, Dhmax should be chosen as small as possible in order to have an accurate temporal evolution. On the other hand, Dhmax should be chosen as large as possible in order to limit the computational time and memory usage. Taking both aspects into account, a Dhmax of 0.05 m has been chosen. [41] The morphodynamic evolution of the bed perturbations is studied by making a Fourier decomposition of the bed perturbation in the longshore direction. The resulting Fourier coefficients are still functions of the cross-shore coordinate. The length of the domain in which the bed perturbation is decomposed in its Fourier components, denoted by LFD, is chosen large enough to represent the linearly most unstable modes, but small enough to be well away from the domain boundaries. The wavelengths that can be represented are LFD/n m, with n = 1, 2, . . ., LFD/20. With n = LFD/20 the smallest wavelength becomes 20 m, which is the Nyquist wavelength since the grid size is 10 m.

Figure 9. Experiment L4: bed and flow perturbations associated with l = 2200 m.

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Table 3. Summarized Settings of the Linear Experiments Experiment

Hs , m

zb2, m

q, degrees

L1 (reference) L2 L3 L4

1.1 3.0 1.1 3.0

2.7 2.7 2.0 2.0

10 10 10 10

The amplitudes obtained in this way are not the amplitudes of the individual eigenmodes of the system. The model under consideration does not allow to get this detailed information, since only the most unstable eigenmode per longshore wavelength is available. Hence the bed perturbation can not be decomposed in terms of its eigenfunctions as was done by, for example, Calvete and de Swart [2003]. [42] To visualize the results, the temporal evolution of the amplitudes of a number of Fourier modes is plotted at two cross-shore positions (x = 160 and 340 m), representative of the behavior at other sections of the bars. Usually, the presented modes concern the four or five modes having the largest amplitude at the end of the experiment. These modes are referred to as the most energetic modes. [43] In the first three nonlinear experiments (experiments NL1, NL2 and NL3; see Table 3) the longshore-uniform bed is perturbed with only one eigenfunction. In experiments NL1 and NL2 the parameter settings of experiments L1 and L4 have been applied, respectively, and the eigenfunctions of the respective FGMs are prescribed as initial bed perturbations. Furthermore, experiment NL1 has been used to assess the influence of the height of the vertical wall at the coastline on the morphodynamic evolution of the coastal system. Experiment NL3 uses the settings of experiment L4 as well, but with the eigenfunction associated with a longshore wavelength of 700 m prescribed as an initial bed perturbation. Note that this wavelength is one of the local maxima in Figure 7 and is about the preferred spacing of experiment L1. In experiment NL4 the settings of experiment L4 are used, but the initial bed perturbation consists of the sum of five eigenfunctions. Experiments NL1, NL2, NL3 and NL4 are discussed in section 5.1. [44] In experiments NL5 (using the parameter settings of L1) and NL6 (with the parameter settings of experiment L4) a random initial bed perturbation has been imposed. Besides, experiment NL5 is used to explore the sensitivity of the morphodynamic evolution to the grid cell size, bed slope related sediment transport and the angle of wave incidence (normal versus oblique). Experiments NL5 and NL6 are presented in section 5.2. The value of LFD in experiments NL1, NL2 and NL3 has been chosen such that the initially imposed wavelength fits exactly in the domain of interest; see also Table 3. 5.1. Nonlinear Experiments With Initial Bed Perturbations Consisting of Eigenfunctions [45] First of all, the results on an aggregated scale are discussed in section 5.1.1. Next, a discussion on a more detailed scale is provided in section 5.1.2. See also Klein [2005] for more details. 5.1.1. Evolution on an Aggregated Scale [46] A coastal system is in equilibrium if no bed changes occur. In the present study, however, the bed forms are migrating due to the oblique wave incidence and therefore

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bed changes will always occur. Therefore we define the coastal system in morphodynamic equilibrium on a more aggregated level if the root-mean-square amplitude (ARMS) of the bed perturbation in the domain in which the Fourier coefficients have been determined does not change significantly in time anymore and if the shapes of the bed forms and the associated migration rates do not significantly vary in time. This still allows for migration of the bed forms and small-scale bed changes. By studying the various modes in detail, it can be assessed whether a morphodynamic equilibrium in the stricter sense is reached as well. For the nonlinear experiments NL1, NL2, NL3 and NL4, ARMS is plotted as a function of time in Figure 10. [47] First of all, Figure 10 illustrates that the root-meansquare amplitude depends on the wave height, since experiments NL2, NL3 and NL4 are performed with a larger wave height than experiment NL1 and generally have a larger ARMS. It can be concluded that the more energetic the hydrodynamic conditions are, the larger the (aggregated) amplitudes of the bed perturbation become. [48] On this aggregated scale and for the duration of the experiments considered, no morphodynamic equilibrium is obtained, contrary to the findings of Damgaard et al. [2002]. Also, in work by Reniers et al. [2004] the coastal system seems to go to an equilibrium although the coastal system is still developing and therefore no definitive conclusions can be drawn. During the amplitude evolution both a linear and a nonlinear phase of growth can be discerned. In the linear phase, ARMS is increasing exponentially in time. This exponential growth levels off and the nonlinear phase begins, in which ARMS still varies significantly in time. Also, the shape of the bed forms and their associated migration still vary significantly in time. After 150 hours of evolution, very large-scale modes emerge which require a longer model domain in order to capture these modes well with the Fourier decomposition. Besides, a large-amplitude perturbation near the coastline emerges. The results after 150 hours therefore have to be used with care.

Figure 10. Root-mean-square amplitude ARMS of the bed perturbation in the domain in which the Fourier coefficients have been determined for the four nonlinear experiments versus time.

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Figure 11. Inner bar of experiment NL1: temporal evolution of the four most energetic modes.

[49] The duration of the phase of initial exponential growth can be determined by considering the natural logarithm of ARMS(t)/ARMS(0). As long as this line is a straight line, the growth of the bed perturbation as a whole is exponential. Applying this to Figure 10 yields durations of 47, 21, 32 and 32 hours in the case of experiments NL1, NL2, NL3 and NL4, respectively. It seems that the development in the low-energy case (experiment NL1) is slower than in the case of the high-energy cases.

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5.1.2. Evolution on a Detailed Scale 5.1.2.1. Experiment NL1 [50] In Figure 11 the amplitude evolution of the four most energetic modes on the inner bar is shown. No significant bed perturbation develops on the outer bar. Until t = 150 hours, the amplitude associated with l = 600 m is much larger than the amplitudes of all other modes and only perturbations with a wavelength smaller than 600 m have a nonnegligible amplitude (not shown in Figure 11). After t = 150 hours, seven modes, all with wavelengths larger than 600 m, have amplitudes of comparable magnitude. Only the first four modes are shown in Figure 11. [51] Plan views of bed perturbations at different moments are depicted in Figures 12 and 13. When only one wavelength is clearly dominating, the bed form can be characterized as a crescentic pattern (left panel of Figure 12), although it is not symmetric anymore. When other modes get a significant amplitude, both up-current (e.g., at t  75 hours; see right panel of Figure 12, when the mode with a wavelength of 300 m has an amplitude comparable to the one with a wavelength of 600 m) and down-current (e.g., at t  142 hours; see left panel of Figure 13) oriented bars, or a mix thereof (e.g., from t = 150 hours on; see right panel of Figure 13, when multiple modes have significant amplitudes) can be observed. [52] Examining the bed patterns around the inner bar after 150 hours, for example, shows that the visually observed length scale of the bed forms can be rather different from the wavelengths of the dominant modes that result in these bed patterns. This is because the total bed form is a sum of these and many other, less significant, modes. Therefore, besides the wavelengths of the most energetic modes, an

Figure 12. Experiment NL1: bed perturbations at (left) t = 55.9 and (right) 74.9 hours. The amplitude of the bed perturbation is in meters. The contour line is the 0.05-m contour. 9 of 19

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Figure 13. Experiment NL1: bed perturbations at (left) t = 142.0 and (right) 193.0 hours. The amplitude of the bed perturbation is in meters. The contour line is the 0.05-m contour.

overall wavelength is presented as well, based on a visual (and hence subjective) estimate of the wavelength that resembles the observed bed pattern best. In the case of experiment NL1, the bed forms have an overall wavelength of the order of 1000 m. The bed forms have an amplitude of about 1.0 m and migrate in downstream direction with a migration celerity of approximately 100 m d
1. [53] To investigate the effects of the height of the vertical wall at the beach, an experiment with the default settings has been performed but now with a vertical wall of 0.1 m instead of 1.0 m. Owing to the relatively steep slope of the bed near the beach (approximately 0.03), this smaller height of the vertical wall implies an increase of the distance between the inner bar crest and the beach with only 30 m. Neither the morphodynamic evolution on an aggregated scale nor the wavelength of the most dominant modes is affected by this larger distance between the inner bar and the ‘‘beach.’’ The main difference is the development of largeamplitude perturbations near the coast, which develop earlier in the case of the 0.1-m-high wall than in the case of the 1.0-m-high wall. The focus in this study is the evolution of bed forms on the breaker bars rather than near the beach. Therefore, without the introduction of a critical velocity for the sediment transport to prevent these unrealistic features, the experiments with a water depth of 1.0 m near the beach are preferred. 5.1.2.2. Experiment NL2 [54] The amplitude evolutions of the four most energetic modes on the inner and the outer bar are shown in Figures 14 and 15, respectively. The two most energetic modes (l = 1100 and 2200 m) on the inner bar dominate the bed forms on this bar. These bed forms have an amplitude of about 0.35 m

and their shape can be characterized as crescentic features throughout the whole considered period. On the outer bar the bed forms evolve from symmetric, crescentic features via asymmetric crescentic features into down-current or undulating bars running across the outer bar crest. The overall wavelength of the bed forms on the outer bar is obviously 2200 m and the average amplitude of the bed feature is 1.35 m. [55] Also in this experiment, the bed perturbations migrate in downstream direction. It is noticeable that the bed pertur-

Figure 14. Inner bar of experiment NL2: temporal evolution of the four most energetic modes.

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Figure 15. Outer bar of experiment NL2: temporal evolution of the four most energetic modes.

Figure 17. Outer bar of experiment NL3: temporal evolution of the five most energetic modes.

bations on the outer bar migrate significantly faster than the ones on the inner bar, namely, 1100 versus 230 m d
1. 5.1.2.3. Experiment NL3 [56] Experiment NL3 is similar to experiment NL2 with the difference that now an initial perturbation associated with a wavelength of 700 m is imposed. The temporal evolution of the most dominant modes on the inner and outer bar are depicted in Figures 16 and 17, respectively. In the evolution of the bed forms on both bars the symmetry of the bed forms breaks, yielding an irregular pattern of crescentic features. On the inner bar the average wavelength of the bed forms is approximately 1000 m. These features have an amplitude of about 0.7 m and migrate downstream with a celerity of 160 m d
1. On the outer bar, the visually observed wavelength of the bed forms is somewhat larger, namely, 1500 m. The averaged amplitude on the outer bar is 1.6 m and the bed forms migrate downstream also with a celerity of 160 m d
1.

5.1.2.4. Experiment NL4 [57] Experiment NL4 concerns a morphodynamic experiment with the settings of experiment L4 in which the initial bed perturbation is the sum of five eigenfunctions, viz. the ones associated with the (local) maxima of Figure 7. Figures 18 and 19 depict the five most energetic modes on the inner and outer bar, respectively. The bed perturbations at t = 56 and 78.6 hours are depicted in Figure 20. The initial bed perturbation quickly develops into an asymmetric crescentic pattern on both bars; see the left panel of Figure 20. This pattern is quite persistent until about t = 70 hours, when no mode is really dominating and many modes have a significant amplitude. The right plot of Figure 20 illustrates that the overall wavelength of the bed perturbation on the inner bar is about 1000 m and the averaged amplitude is 0.5 m. The migration celerity of the bed forms on the inner bar is 220 m d
1. On the outer bar, the apparent wavelength is approximately

Figure 16. Inner bar of experiment NL3: temporal evolution of the five most energetic modes.

Figure 18. Inner bar of experiment NL4: temporal evolution of the five most energetic modes.

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Figure 19. Outer bar of experiment NL4: temporal evolution of the five most energetic modes. 1800 m with an averaged amplitude of 1.4 m and a migration celerity of 270 m d
1. 5.2. Nonlinear Experiments With Random Initial Seeding 5.2.1. Evolution on an Aggregated Scale [58] Two experiments with random initial bed perturbations, imposed on the crests of the two breaker bars, have been performed; see Table 3. To reduce the computation time, the grid cell size has been enlarged to 20 20 m2. No significant differences are found when comparing these results with those of experiments on a finer grid (10

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Figure 21. ARMS versus time of the bed perturbation in the domain in which the Fourier modes have been determined for experiments NL1, NL5B, NL5, and NL6. 10 m2). In experiment NL5, both the complete [Bailard, 1981] sediment transport formulation, including bed slope related sediment transport, and the formulation without the bed slope contribution have been used. The latter experiment is denoted as experiment NL5B. In experiment NL6, the complete [Bailard, 1981] formulation has been used. The settings of these experiments are summarized in Table 3. [59] The temporal evolutions of ARMS of experiments NL1, NL5B, NL5 and NL6 are depicted in Figure 21. Although there are quantitative differences in the growth rate in the linear phase and in the variation of ARMS in the nonlinear phase, the results of experiments NL1, NL5 and

Figure 20. Experiment NL4: bed perturbations at (left) t = 56 hours and at (right) t = 78.6 hours. The amplitude of the bed perturbation is in meters. 12 of 19

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Figure 22. Experiment NL5: temporal evolution of the four most energetic modes on the inner bar. NL5B are not essentially different. The average value of ARMS in the dynamic phase is about equal for these three experiments. Hence it must be concluded that neither the grid cell size, nor the initial bed perturbation, nor the inclusion of bed slope related sediment transport significantly affects the morphodynamic evolution on an aggregated scale under the present topographic and hydrodynamic conditions. [60] As observed in the previous section, the ARMS in the dynamic phase is larger in the case of large wave heights and a high outer bar, compare experiment NL6 with experiments

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NL1, NL5 and NL5B in Figure 21. The investigation into the sensitivity to the initial bed perturbation has been pursued one step farther. Instead of imposing the random initial bed perturbation on the bar crests only, experiment NL5 has been repeated with a random initial perturbation imposed on the whole model domain. The results of this experiment are nearly identical to the results of experiment NL5 itself. This holds for both the aggregated evolution and the development of the individual modes, as described in the next section. 5.2.2. Evolution on a Detailed Scale 5.2.2.1. Experiment NL5 [61] The temporal development of the four most energetic modes on the inner bar is depicted in Figure 22. Although the bed forms are highly variable throughout the whole considered period, they can be characterized as crescentic and undulating features. [62] Figure 23 gives as an example the bed perturbations (left panel) and the total bed level (right panel) at t = 110 hours. The bed perturbations on the inner bar, having a maximum amplitude of about 1 m, migrate with a celerity of approximately 50 m d
1. 5.2.2.2. Experiment NL6 [63] The temporal developments of the five most energetic modes on the inner and outer bar of experiment NL6 are depicted in Figures 24 and 25, respectively. The bed perturbations are, like in experiment NL5, highly dynamic, but they can be characterized as crescentic and undulating patterns nonetheless. The bed perturbations (left panel) and the total bed levels (right panel) at t = 120 hours are depicted as an example in Figure 26. On the inner bar the bed perturbations are crescentic throughout the whole considered period, whereas on the outer bar the bed pertur-

Figure 23. Experiment NL5: (left) bed perturbations and (right) total bed level at t = 110 hours. Amplitudes of the bed perturbation and bed levels are in meters. 13 of 19

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Figure 24. Experiment NL6: temporal evolution of the five most energetic modes on the inner bar. bations evolve via crescentic and undulating patterns towards the irregular pattern shown in Figure 26. [64] On the inner bar of experiment NL6, the bed perturbation, mainly constructed by five modes, has an overall wavelength of 1000 m and an amplitude of 0.3 m. The migration celerity is approximately 50 m d
1. The bed perturbation on the outer bar is mainly constructed by four modes, although six more modes have a significant amplitude as well. The apparent wavelength of the bed forms is 2000 m and the migration celerity is about 200 m d
1. The mean amplitude of the bed perturbations at the crest of the outer bar is about 1.5 m, whereas at the flanks of the outer bar amplitudes up to 2 m are found.

6. Discussion 6.1. Linear Stability Analysis [65] In Table 4 the results of the LSA’s are recapitulated. As can be seen from the ratio Aout/Ain, these experiments can be characterized as experiments in which almost no breaking occurs at the outer bar (experiments L1, L2 and L3) and experiments with significant breaking at the outer bar (experiment L4). 6.1.1. Almost No Wave Breaking at the Outer Bar [66] Since almost no waves break at the outer bar of experiments L1, L2 and L3, it is to be expected that the instability characteristics will be mainly determined by the conditions on the inner bar, and will be similar to those found on a single bar. The stability characteristics of the inner bar have been studied in isolation by performing LSAs over only the interval 70 x < 240 m (see equation (10)) and requiring the bed perturbations to be zero on the outer bar. Both the growth rate and the length scale of the preferred mode are nearly identical to those found in section 4. Hence the growth of a very small perturbation on the outer bar has a negligible contribution to the linear stability characteristics of the whole system. [67] In experiments L1, L2 and L3, the perturbation on the outer bar is mainly formed by the circulations cells in the troughs that are a result of the morphodynamic insta-

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bilities on the inner bar. The longer the wavelength of the perturbation, the larger the horizontal extent of the circulation cell, and hence the larger the relative amplitude one observes at the outer bar. [68] Apart from these observations with respect to experiments L1, L2 and L3, Table 4 indicates that increasing the significant wave height at the seaward boundary from 1.1 m to 3 m for fixed height of the outer breaker bar results in an increase of the preferred spacing from 600 m to 900 m and an increase of the growth rate from 2.84 d
1 to 3.28 d
1; see Figure 6 as well. These results correspond with the findings of Deigaard et al. [1999], Calvete et al. [2002] and Coco et al. [2002]. Increasing the height of the outer breaker bar while fixing the wave height at 1.1 m (experiment L3) results in a smaller preferred spacing and smaller growth rate. [69] Increasing the wave height at the inner bar results in an increase of both the mean longshore current velocity and the mean significant wave height. Increasing the outer bar height, the rate of breaking at the outer bar increases and consequently a smaller mean longshore current velocity and a smaller mean significant wave height at the inner breaker bar are found. To determine whether the longshore current velocity or the wave height (via the wave orbital velocity) determines the preferred spacing and the associated growth rate, experiments in which either the longshore current velocity at the inner bar is changed while keeping the wave height fixed, or the wave height is varied and the longshore current profile is fixed, have to be performed. Unfortunately, it is very hard to perform such an experiment for a doublebarred beach, but they can be carried out for a single-barred beach. Since the linear stability characteristics of the doublebarred beach of experiment L1 are rather insensitive to the presence of the outer bar, the results obtained with the single-barred analogous of experiment L1 can be used to explain the observations found when studying a doublebarred beach geometry. Experiment L1A is the single-barred analogous of experiment L1 by setting zb2 and zt2 to zero, whereas all other parameters have default values. The settings of experiment L1B are identical to those of exper-

Figure 25. Experiment NL6: temporal evolution of the five most energetic modes on the outer bar.

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Figure 26. Experiment NL6: (left) bed perturbations and (right) total bed level at t = 120 hours. Amplitudes of the bed perturbation and bed levels are in meters. iment L1A except for a doubling of the significant wave height to 2.2 m and a friction coefficient that has been adjusted such that the maximum mean longshore current velocity at the inner bar is the same as in experiment L1A. In Figure 27 the growth rates of experiments L1, L1A and L1B are shown. [70] Increasing the wave height and the drag coefficient in the single inner bar experiment yields a preferred spacing of 500 m, which is 100 m smaller than the preferred spacing of experiments L1 and L1A. The growth rate is significantly larger than the one of experiment L1A, namely, 4.40 d
1 instead of 2.96 d
1. Although some differences in the longshore current velocity profile on the seaward slope of the inner bar exist, the overall current velocity profile is quite similar to the one obtained with experiment L1A. Hence the difference in the growth rate must be attributed to the larger wave height at the inner bar, either by an increased sediment stirring or due to transport in the direction of the wave orbital motion. [71] The wavelengths in experiments L1, L1A and L1B are similar, which implies that a change in the wave height does not result in an essentially different preferred spacing. From experiments L1, L2 and L3 it can be concluded that an increase of the magnitude of the longshore current yields

an increase of the preferred spacing (Figure 6). This dependence of the preferred spacing on the magnitude of the longshore current velocity agree with the explanation given by Deigaard et al. [1999] who state that the preferred spacing increases with increasing inertia of the longshore current. This means that a stronger longshore current or a larger trough volume yield larger spacings. [72] To analyze in more detail how the wave height influences the growth rate, the sediment transport vector is decomposed in bed and suspended load contributions in the directions of the wave orbital motion and the mean current.

Table 4. Summarized Results of the Linear Experiments Exp.

Hs, m

zb2, m

q, degrees

lp, m

wi,p, d
1

wr,p, d
1

cp, m d
1

Aout/Ain, %

L1 L2 L3 L4

1.1 3.0 1.1 3.0

2.7 2.7 2.0 2.0

10 10 10 10

600 900 500 2200

2.84 3.28 1.78 3.24

0.83 1.73 0.05 1.10

79.6 247.8 4.0 385.2

1.3 10.2 1.6 594.3

Figure 27. Experiments L1, L1A, and L1B: growth rate versus wavelength.

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Figure 28. Growth rate as a function of the longshore wavelength in case of the LSA of the whole system, the LSA of the single inner bar, and the LSA of the single outer bar, obtained with the settings of experiment L4. An analysis of these terms obtained with experiments L1, L2 and L3 shows that cross-shore suspended sediment transport in the current direction is at least two times larger than the second largest contribution, which is cross-shore bed load transport in the direction of the wave orbital motion. Therefore the sediment transport in leading order can be estimated as a(x, y)uc, with a(x, y) the sediment stirring function and uc the cross-shore current velocity. Assuming that this stirring function can be well-approximated by a(x), i.e., a stirring function which is averaged in the longshore direction, the analysis of Falque´s et al. [1996] can be followed. They show that whether a bed perturbation grows or not depends on the direction of the cross-shore current and the cross-shore gradient of the ratio of the sediment stirring function and the reference depth [see also Ribas et al., 2003; Klein and Schuttelaars, 2005]. For this analysis they use the so-called bottom evolution equation (BEE),
 @h0 aV0 @h0 @ a 0 u: þ ¼
a ln @t D0 @y @x D0 c

ð11Þ

Here D0 is the equilibrium water depth, V0 the mean longshore current velocity and a is the sediment stirring function, obtained by averaging a(x, y) in longshore direction. The second term on the left-hand side only results in migration, whereas the term on the right-hand side can result in growth of bed forms. The function a/D0 is called the potential stirring function. [73] Comparison of the terms of the BEE, applied to experiments L1A and L1B, shows that the sediment stirring function is significantly affected by the larger wave height at the inner bar. The relative change of the sediment stirring function is much larger than the relative change of the crossshore current velocity. Since the sediment transport contributions in the direction of the wave orbital motion are for both experiments approximately the same, the relatively large growth rate of experiment L1B can most probably be

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attributed to an increased rate of sediment stirring due to the larger wave height. 6.1.2. Significant Wave Breaking at the Outer Bar [74] Multiple (local) maxima have been found with the LSA of experiment L4. In order to identify the origin of these maxima, single inner bar (i.e., omitting the outer breaker bar) and single outer bar (i.e., omitting the inner breaker bar) analogues of experiment L4 have been performed. The results of these LSAs, together with the results of experiment L4, are depicted in Figure 28. [75] The LSAs of the single inner bar and the single outer bar yield a single maximum at l = 1000 and 2100 m, respectively. The associated growth rates are larger than the growth rates obtained with the LSA of the whole system. In the case of the single inner bar the omission of the outer bar results in larger waves at the inner bar and consequently in a larger growth rate. For the single outer bar case, however, the cause of the larger growth rate is less obvious. The hydrodynamics around the outer bar hardly change when the inner bar is omitted. Therefore it can be inferred that the development of bed perturbations on the inner bar have a damping effect on the growth rate of the whole system. Since the other two local maxima at l = 400 and 700 m are not found in the single bar experiments, it must be concluded that these modes originate from hydrodynamic interactions between the inner and outer bar. [76] Hence, in the case of significant wave breaking on the outer bar, the linear stability characteristics are determined by a complex hydro- and morphodynamic interaction between the two bars. This hydrodynamic feedback consists of circulation cells resulting from instabilities on one bar interacting with the morphodynamic instabilities on the other bar, and of longshore nonuniform waves reaching the inner bar owing to the bed perturbations on the outer bar. 6.2. Nonlinear Experiments [77] The results of the nonlinear experiments have been summarized in Table 5. In all experiments the ARMS finally displays small-scale variations superposed on a slow linear increase in the dynamic phase. This is especially clear in experiment NL1. The linear increase can be attributed to the emergence of modes with wavelengths larger than 3000 m. These large-scale modes are not captured very well since the domain in which the longshore Fourier components of the bed perturbation are computed is ‘‘only’’ 6000 m. Therefore the experiments cannot be extended too far in time without increasing the size of the model domain. Apart from that, a large-amplitude bed perturbation emerges at some locations near the coast, even though the depth there has been set to Table 5. Summarized Results of the Nonlinear Experiments: Wavelength l, Migration Celerity c, and Amplitude A of the Bed Forms, and Duration TL of the Phase of Linear Growth Inner Bar Exp. TL, hours NL1 NL2 NL3 NL4 NL5 NL6

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47 21 32 32 70 60

l, m 1000 1100 and 2200 1000 1000 1000 1000

Outer Bar
1

c, m d 100 230 160 220 50 50

A, m l, m c, m d
1 A, m 1.0 0.35 0.7 0.5 1.0 0.3

2200 1500 1800

1100 160 270

1.35 1.6 1.4

2000

200

1.5

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1 m. Therefore the experiments were stopped after a certain amount of time, depending on the moment at which the large-scale modes and the perturbation near the coastline emerge. [78] It can be concluded that in the case of the low-energy experiment (NL1) the visually observed wavelength of 600 m corresponds during a long period very well with the wavelength of the FGM. In the second stage of the nonlinear phase, the overall wavelength becomes of the order of 1000 m. The migration rate in the nonlinear phase is quite similar to the one found with the LSA. The bed perturbation, on the contrary, is highly variable and therefore resembles the spatial structure of the FGM only occasionally. [79] It is clear that in all nonlinear experiments with the settings of experiment L4, different spacings prevail on either bar, namely, 1000 and 2000 m on the inner and outer bar, respectively. The LSA only finds one preferred spacing for the whole system and hence is not able to predict the wavelengths on the bars correctly. Besides, the migration celerities of the bed forms on the inner and outer bar differ significantly. [80] In all six nonlinear experiments done in this study, a highly dynamic morphological behavior of the bed forms was observed and no final equilibrium solution was reached, contrary to the findings of Damgaard et al. [2002]. Here we define a coastal system to be morphologically in equilibrium if (1) ARMS does not change significantly in time, (2) the shape of the bed forms does not change in time, and (3) the migration celerity of the bed forms is constant. Although the variation of ARMS after t = 150 hours is not large, the bed forms and their migration rate are still varying in time. It is not likely that an equilibrium will be found if the experiments would have been pursued longer (in a larger model domain and with the introduction of a critical velocity to prevent the large amplitude perturbation near the coast). In work by Damgaard et al. [2002] the coastal system reached an equilibrium solution after about 50 days, which is approximately twice their e-folding time, whereas the nonlinear experiments in present paper have been pursued for at least 3 times the duration of the phase of initial growth. Apart from that, the behavior of both the individual modes and the evolution on an aggregated scale, especially the shapes of the bed forms and the migration rates, vary significantly in time. [81] There are five major differences between the present model and the model of Damgaard et al. [2002], which are potential reasons for the difference in morphodynamic behavior, namely, (1) the number of breaker bars, (2) bed slope related sediment transport, (3) application of a critical velocity for sediment transport, (4) the angle of wave incidence and (5) the sediment transport formulation. [82] Damgaard et al. [2002] considered single-barred beaches whereas in the present paper double-barred beaches have been considered. Since experiment NL1, which has been shown to behave as a single barred beach, exhibits continuous dynamic behavior as well, the difference in the number of bars can not be responsible for the different behavior. [83] In section 5.2 it has already been shown that the implementation of bed slope effects does not essentially change the morphodynamic evolution and leads not to a morphodynamic equilibrium.

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[84] The application of a critical velocity is of minor importance, since the current velocities on top of the breaker bars, where all the changes take place, are much larger than reasonable values of the critical velocity. Hence the application of a critical velocity does not prohibit morphological changes. Also the reduction of the current velocity (by subtracting the critical velocity) is small and does therefore not essentially affect the sediment transport rates. Note, however, that application of a critical velocity would probably prevent the emergence of large-amplitude perturbations near the coastline. [85] Setting the angle of wave incidence to zero also does not yield an equilibrium state, since ARMS of an experiment with normal wave incidence (experiment not shown) continues to vary in time. Therefore it must be concluded that it is the sediment transport formulation that is responsible for the difference in behavior found in the present paper and in work by Damgaard et al. [2002]. An experiment with the same sediment transport formulation as the one by Damgaard et al. [2002] is beyond the scope of this paper. However, an additional experiment with only bed load transport in the current direction (similar as has been done by Klein and Schuttelaars [2005]) shows that it is not only the sediment transport in the direction of the wave orbital velocity that causes the dynamic behavior, although the variation in ARMS has decreased. 6.3. Comparison With Observations [86] As Table 5 shows, the inner bar bed forms with a wavelength of approximately 1000 m and a migration celerity ranging between 100 and 230 m d
1 are found. The bed forms on the inner bar are highly variable and can be characterized as crescentic and undulating bed features. On the outer bar much longer bed forms are found, with wavelengths ranging between 1500 and 2200 m. The bed forms, which can be characterized as undulating features, migrate downstream with a celerity ranging between 160 and 1100 m d
1. [87] On the basis of a 6-week data set observed at Egmond, Ruessink et al. [2000] showed that 85% of the variability of the inner bar crest was related to horizontal amplitude variations and longshore migration of bed features with a longshore wavelength of 600 m. The migration rate of these bed features varied between 0 – 150 m d
1, depending on the longshore component of the wave power (see Ruessink et al. [2000] for a definition). In addition, Wijnberg [1995] observed bed features with much larger wavelengths, ranging between 1000 and 3000 m on the outer bar of Egmond. [88] On the basis of a 3.4-year-long data set of daily video images of the surf zone of Noordwijk, having a crossshore profile comparable with Egmond, Van Enckevort and Ruessink [2003] derived characteristic bed forms and associated wavelengths, migration rates and cross-shore amplitudes. They found a wide variety of bed forms on the inner bar, ranging from small-scale rip channel systems (averaged l = 430 m), crescentic bars (averaged l = 990 m), irregular features (averaged l = 1850 m) and undulating bed forms (averaged l = 2500 m). On the outer bar, mainly the undulating and crescentic features with the same wavelengths have been observed. In addition to these observations, Short [1992] observed from aerial photographs bed

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features on the inner bar with wavelengths between 350 and 900 m. Ruessink et al. [2000] and Van Enckevort and Ruessink [2003] found the migration rate to depend on the longshore component of the wave power, as a proxy for magnitude of the longshore current. It varied between 0 and 180 m d
1, and the migration rate is somewhat larger on the outer bar than on the inner bar. [89] In general, it can be concluded that the spacings obtained in the nonlinear experiments correspond well with the observations made by Short [1992], Wijnberg [1995], Ruessink et al. [2000] and Van Enckevort and Ruessink [2003]. [90] Like the observations, the morphodynamic experiments discussed in this paper yield a migration rate on the outer bar that is generally larger than the one on the inner bar. For the low-energy experiment NL1, the migration rate obtained corresponds very well with observations. In the high-energy cases, the migration rates are larger than but of the same order of magnitude as the ones observed. [91] A comparison of the bed perturbation resulting from the LSAs and the final bed perturbations resulting from the morphodynamic experiments shows that the bed perturbations are no longer symmetric when nonlinear effects become important. In the nonlinear experiments, the shoals become more horn-like whereas the pools become more elongated. Furthermore, the height of the shoals is larger than the depth of the pools. Besides these crescentic bed features, the nonlinear experiments also found up- and down-current (undulating) bars and irregular patterns. The shape of crescentic bars corresponds rather well with the description of crescentic bars given by, for example, Komar [1976] and Van Enckevort et al. [2004]. Furthermore, the variety in shapes of the bed forms and their characteristics are also described in the studies mentioned. [92] Although the classification of bed forms in the surf zone of, for example, Wright and Short [1984] is discrete, Van Enckevort and Ruessink [2003] show that the transitions from one bed form and apparent spacing to another are gradual. No simple, linear relation between the transitions and changes in the hydrodynamic forcing could be distinguished. This corresponds very well with the findings of the present paper (section 5), namely, the morphodynamic behavior of coastal systems is highly dynamic, even with constant forcing.

7. Conclusions [93] The stability of double-barred beaches in the linear and the nonlinear regime has been studied using a fully nonlinear numerical model. The linear stability analyses resulted in bed perturbations that can be characterized as crescentic bed patterns. It has been shown that the magnitude of the longshore current is important for the preferred spacing whereas the wave height, via the sediment stirring, is important for the growth rate. Whether a bed perturbation develops on the outer bar depends on the height of the bar, the wave height and the longshore wavelength of the bed perturbation. [94] The nonlinear, morphodynamic behavior of the double-barred coastal systems under consideration can be characterized by a phase of exponential growth and, consecutively, a phase of dynamic behavior. After approxi-

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mately 150 hours, large-scale modes are emerging and large-amplitude perturbations near the coast are developing. Therefore the results after approximately t = 170 hours should be used with care. Nonetheless, it can be concluded that the coastal system is, according to our definition (see section 5.1.1), not in morphodynamic equilibrium, since both the root-mean-square amplitude of the bed perturbations and the shape of the bed forms and associated migration rates are still significantly varying in time. [95] Furthermore, it appears that the dynamic behavior of bed forms on the outer bar depends on the initial bed perturbation (see experiment NL2 and experiments NL1, NL3 and NL4), which corresponds with the findings of Damgaard et al. [2002]. The longer the wavelength of the initial bed perturbation, the longer the overall spacing and the faster the migration celerity of the emerging bed forms. On the inner bar, however, the wavelength and the celerity of the resulting bed forms are quite insensitive to the initial bed perturbation. The visually observed wavelength of the bed forms on the inner bar is in all experiments about 1000 m. Note that the wavelengths of the individual modes, spanning this bed form, differ per experiment, depending on the initial perturbation. [96] The preferred spacing and associated migration rates as found with the LSAs are only a good prediction of the apparent spacing and migration rates in the first 40 to 60 hours of the nonlinear experiments, i.e., the phase of exponential growth. In the dynamic phase, apart from the FGM, other modes contribute significantly to the bed form, causing a visually observed length scale of the bed forms which differs from the wavelengths of the dominant modes. The migration rates found in the linear analysis are larger than those found in nonlinear experiments. The shapes of the bed perturbations, their highly dynamic character and the apparent wavelengths correspond well with observations. Also, the migration rate is of the same order of magnitude as the migration rates found in the literature. [97] The fact that contrary to Damgaard et al. [2002], no equilibrium is found, must be attributed to the exact formulation of the sediment transport. An experiment with a modified sediment transport formulation has shown that the results of the nonlinear experiments are sensitive to the sediment transport formulation, however not to the degree that has been found by Klein and Schuttelaars [2005] for planar beaches. Also, with the adjusted formulation, no morphodynamic equilibrium has been found. Two arguments substantiate the conclusion that no equilibrium will be found. The first is the fact that the experiments have been performed for at least 3 times the duration of the phase of initial growth, whereas Damgaard et al. [2002] found an equilibrium after already 2 times the e-folding time. The second argument is the fact that both the individual modes and the evolution on an aggregated scale are, after t = 150 hours, still highly dynamic, owing to nonlinear interactions of the individual modes. [98] Acknowledgments. The research presented herein is done in the frame work of the DIOC-programme ‘‘Water’’ of Delft University of Technology. The authors wish to thank Johan van der Molen for kindly making his sediment transport routines available and Ad Reniers for his useful comments. The authors also wish to thank the two anonymous reviewers for their constructive comments.

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M. D. Klein, Svasˇek Hydraulics, P.O. Box 91, NL-3000 AB Rotterdam, Netherlands. ([email protected]) H. M. Schuttelaars, Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, NL-2600 GA Delft, Netherlands. ([email protected])

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